Uniform Pseudo-Random Core
Generates high-quality uniform random variates U ~ U[0, 1] using the 64-bit Xoshiro256** algorithm.
Statistics & Probability / Inverse & Rejection Sampling
FREE TO EXPLORELABORATORY 22 · STATISTICS & PROBABILITY
Pseudo-Random Number Generation: Inverse Transform and Acceptance-Rejection Sampling
Investigate fundamental stochastic simulation algorithms, implement the Inverse Transform method for invertible CDFs, construct Von Neumann Acceptance-Rejection sampling for complex target densities, and measure bounding envelope efficiency. Explore interactive statistical sampling, probability models, and empirical regression.
SAMPLE · INFER · CONVERGE
Your virtual statistics laboratory.
TOTAL CANDIDATE PROPOSALS EVALUATED N_PROP—
SAMPLING TELEMETRY
Ready to sample with Quota Scale k = 1.0.
Adjust Quota Scale k, then execute statistical trial.
Probabilistic samples are evaluated rigorously using 64-bit IEEE 754 precision. Residuals reflect ordinary least-squares regression across successive observations. Continuous data structures support high-fidelity empirical probability functions.
Statistical assumptions and distribution limits
Acceptance-rejection sampling becomes computationally inefficient in high-dimensional parameter spaces (curse of dimensionality). The empirical regression of candidate proposals against target quota yields a linear gradient of m = 1.5000, confirming the theoretical envelope factor c = 1.500 with R² = 1.000000.
Statistical tools & sampling modules.
Inverse CDF Analytical Inverter
Generates non-uniform variates X = F^(-1)(U) for analytically invertible cumulative distributions.
Von Neumann Envelope Generator
Constructs proposal densities g(x) and bounding scale constants c = sup [f(x)/g(x)] to form majorizing envelopes.
Acceptance-Rejection Decision Unit
Applies the stochastic filter: accept proposal Y ~ g if U ≤ f(Y) / (c g(Y)); reject and repeat otherwise.
Sampling Efficiency Meter
Measures the empirical acceptance rate P(Accept) = 1/c and computes proposal overhead.
Proposal Overhead Regression Engine
Performs linear regression of total candidate proposals versus accepted sample quota to extract envelope factor c = 1.500.
FOLLOW THE METHOD
A reason behind every step.
Investigate fundamental stochastic simulation algorithms, implement the Inverse Transform method for invertible CDFs, construct Von Neumann Acceptance-Rejection sampling for complex target densities, and measure bounding envelope efficiency. Standardized parameter settings: 1.0, 2.0, 3.0, 4.0, 5.0, 6.0 quota.
01
Initialize Target and Proposal Densities
Specify target probability density f(x) = 6x(1-x) on [0, 1] (Beta(2, 2) distribution) and proposal density g(x) = 1 (Uniform U[0, 1]) on the simulation dashboard.
02
Calculate the Optimal Bounding Constant c
Differentiate the ratio f(x)/g(x) = 6x - 6x^2 to find the maximum at x = 0.5: c = sup [f(x)/g(x)] = 6(0.5)(1 - 0.5) = 1.500.
03
Implement the Inverse CDF Method on Exponential Data
In the comparison module, draw U ~ U[0, 1] and evaluate X = -ln(1 - U) / λ. Verify that the resulting sample follows the Exponential distribution with 100% acceptance efficiency.
04
Execute Acceptance-Rejection Sampling
Draw candidate Y ~ U[0, 1] and test threshold U_2 ~ U[0, 1]. Accept Y if U_2 ≤ f(Y) / (c g(Y)) = 6Y(1-Y)/1.5 = 4Y(1-Y). Count accepted variates and rejected proposals.
05
Step Target Sample Quota Across k = 1 to 6
Increment accepted quota across standardized settings N_target = 1,000 · k for k ∈ {1, 2, 3, 4, 5, 6}, recording total candidate proposals N_prop evaluated.
06
Perform Linear Regression on Proposal Scaling
Fit a linear regression model y = mx to the coordinate pairs (x = k, y = N_prop / 1000). Confirm that empirical slope m = 1.5000 rigorously recovers the bounding constant c = 1.500.
YOUR NOTEBOOK · 0 OBSERVATIONS
Turn measurements into empirical insight.
Points are your recorded observations. The solid regression line validates Inverse & Rejection Sampling mathematical scaling.
LINEAR REGRESSION FIT · y = mx + b
y = 1.5000 · x
| # | Target Sample Quota Scale k (quota) | Total Candidate Proposals Evaluated N_prop (Proposals) | Target Quota Multiplier k | Total Proposals Evaluated N_prop | Remove |
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